Implementing the three-particle quantization condition including higher partial waves

Abstract

We present an implementation of the relativistic three-particle quantization condition including both sand d-wave two-particle channels. For this, we develop a systematic expansion of the three-particle K matrix, K df,3 , about threshold, which is the generalization of the effective range expansion of the two-particle K matrix, K 2 . Relativistic invariance plays an important role in this expansion. We find that d-wave two-particle channels enter first at quadratic order. We explain how to implement the resulting multichannel quantization condition, and present several examples of its application. We derive the leading dependence of the threshold three-particle state on the two-particle d-wave scattering amplitude, and use this to test our implementation. We show how strong twoparticle d-wave interactions can lead to significant effects on the finite-volume three-particle spectrum, including the possibility of a generalized three-particle Efimov-like bound state. We also explore the application to the 3π + system, which is accessible to lattice QCD simulations, where we study the sensitivity of the spectrum to the components of K df,3 . Finally, we investigate the circumstances under which the quantization condition has unphysical solutions.F Properties of the isotropic approximation 50 G Failure of Eq. (4.34) for quadratic and cubic terms in the threshold expansion 53There is also an induced three-particle interaction due to the exchange of a virtual particle between a pair of two-particle interactions. This is included in all approaches.2 The p wave is absent due to Bose symmetry. 3 It is expected, however, that there is no barrier to extending to higher waves.-3 -Z 2 symmetry that forbids 2 ↔ 3 transitions. Another important feature of this approach is that it can be made relativistic [5], which turns out to simplify the expansion about threshold. Although we use the RFT approach, we expect that many of the technical considerations and general conclusions will apply to all three approaches to the quantization condition.The formalism of Ref.[1] is restricted to two-particle interactions that do not lead to poles in K 2 , the two-particle K matrix. If there are such poles, then one should use the generalized, and more complicated, formalism derived in Ref. [7]. For simplicity, we consider here only examples in which there are no K-matrix poles.Since our main goal is to show how the formalism works when including higher waves, our numerical examples are mainly chosen for illustrative purposes and do not represent physical systems. However, there is one case in nature for which our simplified setting applies, namely the 3π + system. Thus, in one of our examples, we set the two-particle scattering parameters to those measured experimentally for two charged pions, and illustrate the dependence of the resulting three-pion spectrum on the three-particle scattering parameters. This is similar to the study made in Ref.[15] using the FVU approach, except here we include d-wave dimers.All three-particle quantization condi...

I=3 Three-Pion Scattering Amplitude from Lattice QCD

“…One way to see that this is indeed the case is to use the threshold expansion developed in Ref. [38]. At any order in the expansion parameter ∆, one can show that the only terms that are purely s-wave are those that are isotropic.…”

“…The procedure for decomposing the QC in this case was first presented in Ref. [38], but the generalization to arbitrary P is new to this work and deserves explanation. Unlike in Ref.…”

“…Unlike in Ref. [38] where both max ∈ {0, 2} were considered, here we focus only on the relevant case of max = 0.…”

“…We therefore seek to decompose the QC into irreps of LG( P ) instead of O h , but otherwise the recipe used in Ref. [38] is unchanged from that in Ref. [38].…”

Three-body unitarity versus finite-volume π+π+π+ spectrum from lattice QCD

“…[38]; higher-spin twoparticle sub-systems were considered in Ref. [57]. First numerical studies [4,35,37] demonstrated the feasibility of different formalisms.…”